Random Matrices

569 Pages·1990·21.586 MB·English

by  Madan Lal Mehta (Auth.)

Random
Matrices
REVISED AND ENLARGED
SECOND EDITION
MADAN LAL MEHTA
Centre d'Etudes Nucléaires de Saclay
Gif-sur-Yvette Cedex
France
Centre National de Recherche Scientifique
France
ACADEMIC PRESS, INC.
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Library of Congress Cataloging-in-Publication Data
Mehta, M. L.
Random matrices / Madan Lai Mehta.—Rev. and enl 2nd ed.
p. cm.
Includes bibliographical references.
ISBN 0-12-488051-7 (alk. paper)
1. Energy levels (Quantum mechanics)—Statistical methods.
2. Random matrices. I. Title.
QC174.45.M444 1990 90-257
530.l'2—dc20 CIP
Printed in the United States of America
90 91 92 93 9 8 7 6 5 4 3 2 1
Preface to the Second Edition
The contemporary textbooks on classical or quantum mechanics deal
with systems governed by differential equations that are simple enough to
be solved in closed terms (or eventually perturbatively). Hence, the entire
past and future of such systems can be deduced from a knowledge of their
present state (initial conditions). Moreover, these solutions are stable in the
sense that small changes in the initial conditions result in small changes
in their time evolution. Such systems are called integrable. Physicists and
mathematicians now realize that most of the systems in nature are not
integrable. The forces and interactions are so complicated that either we
cannot write the corresponding differential equation, or when we can, the
whole situation is unstable: a small change in the initial conditions produces
a large difference in the final outcome. They are called chaotic. The relation
of chaotic to integrable systems is something like that of transcendental to
rational numbers.
For chaotic systems, it is meaningless to calculate the future evolution
starting from an exactly given present state, because a small error or change
at the beginning will make the whole computation useless. One should
rather try to determine the statistical properties of such systems.
The theory of random matrices makes the hypothesis that the charac-
teristic energies of chaotic systems behave locally as if they were the eigen-
values of a matrix with randomly distributed elements. Random matrices
were first encountered in mathematical statistics by Hsu, Wishart, and oth-
ers in the 1930s, but an intensive study of their properties in connection
with nuclear physics only began with the work of Wigner in the 1950s. In
1965, C. E. Porter edited a reprint volume of all important papers on the
subject, with a critical and detailed introduction, that even today is very
instructive. The first edition of the present book appeared in 1967. Dur-
ing the last two decades there have been many new results, and a larger
number of physicists and mathematicians became interested in the subject
xi
xii Preface to the Second Edition
owing to various potential applications. Consequently, it was felt that this
book had to be revised even though a nice review article by Brody et al.
has appeared in the meantime (Rev. Mod. Phys., 1981).
Among the important new results, one notes the theory of matrices with
quaternion elements, which serves to compute some multiple integrals, the
evaluation of r^point spacing probabilities, the derivation of the asymptotic
behavior of nearest neighbor spacings, the computation of a few hundred
millions of zeros of the Riemann zeta function and the analysis of their
statistical properties, the rediscovery of Selberg's 1944 paper, giving rise
to hundreds of recent publications, the use of the diffusion equation to
evaluate an integral over the unitary group, thus allowing the analysis of
non-invariant Gaussian ensembles, and the numerical investigation of var-
ious systems with deterministic chaos.
After a brief survey of the symmetry requirements, the Gaussian en-
sembles of random Hermitian matrices are introduced in Chapter 2. In
Chapter 3 the joint probability density of the eigenvalues of such matrices
is derived. In Chapter 5 we give a detailed treatment of the simplest of the
matrix ensembles, the Gaussian unitary one, deriving the n-point correla-
tion functions and the 7>point spacing probabilities. Here we explain how
the Fredholm theory of integral equations can be used to derive the lim-
its of large determinants. In Chapter 6 we study the Gaussian orthogonal
ensemble, which in most cases is appropriate for applications but is math-
ematically more complicated. Here we introduce matrices with quaternion
elements and their determinants as well as the method of integration over
alternate variables. The short Chapter 8 introduces a Brownian motion
model of Gaussian Hermitian matrices. Chapters 9, 10, and 11 deal with
ensembles of unitary random matrices, the mathematical methods being
the same as in Chapters 5 and 6. In Chapter 12 we derive the asymptotic
series for the nearest neighbor spacing probability. In Chapter 14 we study
a non-invariant Gaussian Hermitian ensemble, deriving its 7vpoint correla-
tion and cluster functions; it is a good example of the use of mathematical
tools developed in Chapters 5 and 6. Chapter 16 describes a number of
statistical quantities useful for the analysis of experimental data. Chap-
ter 17 gives a detailed account of Selberg's integral and its consequences.
Other chapters deal with questions or ensembles less important either for
applications or for the mathematical methods used. Numerous appendices
treat secondary mathematical questions, list power series expansions, and
present numerical tables of various functions useful in applications.
Preface to the Second Edition xiii
The methods explained in Chapters 5 and 6 are basic; they are necessary
to understand most of the material presented here. However, Chapter 17
is independent. Chapter 12 is the most difficult one, since it uses results
from the asymptotic analysis of differential equations, Toeplitz determi-
nants, and the inverse scattering theory, for which, in spite of a few nice
references, we are unaware of a royal road. The rest of the material is
self-contained and hopefully quite accessible to any diligent reader with
modest mathematical background. Contrary to the general tendency these
days, this book contains no exercises.
Acknowledgments
By way of education and inspiration I owe much to my late father, P.
C. Mehta, to my late Professor, C. Bloch, and to Professors E. P. Wigner
and F. J. Dyson. I had free access to published and unpublished works of
Wigner, Dyson, and my colleagues M. Gaudin, R. A. Askey, A. Pandey, and
J. des Cloizeaux. I had the privilege to attend the lectures on zeta functions
by H. Cohen, H. Stark, and D. Zagier at Les Houches in March 1989. E. A.
van Doom and A. Gervois read the manuscript separately almost entirely
while R. A. Askey, H. L. Montgomery, and G. Mahoux read portions of it,
thus helping me to avoid black spots and inaccuracies. It is my misfortune
that trying to do a complete job, A. Pandey gave me his comments too
late. O. Bohigas kindly supplied me with a number of graphs. A. Odlyzko
sent me his preprint on the zeros of the Riemann zeta. It is my pleasant
duty to thank all of them.
In contrast to the first edition, this edition contains a large umber of
figures, especially in Chapters 1, 5, 6, and 16. Encouraged by my colleague,
O. Bohigas, this became possible by the generous permissions of various
authors and scientific publications. It is my pleasant duty to thank all of
them.
The major part of the manuscript was typed on an IBM-PC by the
author himself who used the very convenient mathematical word processor
"MATHOR" and then translated it into TEX using the same processor. I
hope that no serious errors persist.
XV
Preface to the First Edition
Though random matrices were first encountered in mathematical statis-
tics by Hsu, Wishart, and others, intensive study of their properties in
connection with nuclear physics began with the work of Wigner in the
1950s. Much material has accumulated since then, and it was felt that it
should be collected. A reprint volume to satisfy this need had been edited
by C. E. Porter with a critical introduction (see References); nevertheless,
the feeling was that a book containing a coherent treatment of the subject
would be welcome.
We make the assumption that the local statistical behavior of the energy
levels of a sufficiently complicated system is simulated by that of the eigen-
values of a random matrix. Chapter 1 is a rapid survey of our understanding
of nuclear spectra from this point of view. The discussion is rather general,
in sharp contrast to the precise problems treated in the rest of the book.
In Chapter 2 an analysis of the usual symmetries that a quantum system
might possess is carried out, and the joint probability density function for
the various matrix elements of the Hamiltonian is derived as a consequence.
The transition from matrix elements to eigenvalues is made in Chapter 3,
and the standard arguments of classical statistical mechanics are applied in
Chapter 4 to derive the eigenvalue density. An unproved conjecture is also
stated. In Chapter 5 the method of integration over alternate variables is
presented, and an application of the Fredholm theory of integral equations
is made to the problem of eigenvalue spacings. The methods developed in
Chapter 5 are basic to an understanding of most of the remaining chapters.
Chapter 6 deals with the correlations and spacings for less useful cases. A
Brownian motion model is described in Chapter 7. Chapters 8 to 11 treat
circular ensembles; Chapters 8 to 10 repeat calculations analogous to those
of Chapters 4 to 7. The integration method discussed in Chapter 11 orig-
inated with Wigner and is being published here for the first time. The
theory of non-Hermitian random matrices, though not applicable to any
xvii
xviii Preface to the First Edition
physical problems, is a fascinating subject and must be studied for its own
sake. In this direction an impressive effort by Ginibre is described in Chap-
ter 12. For the Gaussian ensembles the level density in regions where it is
very low is discussed in Chapter 13. The investigations of Chapter 16 and
Appendices A.29 and A.30 were recently carried out in collaboration with
Professor Wigner at Princeton University. Chapters 14, 15, and 17 treat a
number of other topics. Most of the material in the appendices is either well
known or was published elsewhere and is collected here for ready reference.
It was surprisingly difficult to obtain the proof contained in A.21, while
A.29, A.30, and A.31 are new.
October, 1967 M. L. MEHTA
Saclay, France
1 / Introduction
In the theory of random matrices one is concerned with the following
question. Consider a large matrix whose elements are random variables
with given probability laws. Then what can one say about the proba-
bilities of a few of its eigenvalues or of a few of its eigenvectors? This
question is of pertinence for the understanding of the statistical behavior
of slow neutron resonances in nuclear physics, where it was first proposed
and intensively studied. The question gained importance in other areas
of physics and mathematics, such as the characterization of chaotic sys-
tems, elastodynamic properties of structural materials, conductivity in
disordered metals, or the distribution of the zeros of the Riemann zeta
function. The reasons of this pertinence are not yet clear. The impres-
sion is that some sort of a law of large numbers is in the background. In
this chapter we will try to give reasons why one should study random
matrices.
1.1. Random Matrices in Nuclear Physics
Figure 1.1 shows a typical graph of slow neutron resonances. There one
sees various peaks with different widths and heights located at various
places. Do they have any definite statistical pattern? The locations of the
peaks are called nuclear energy levels, their widths the neutron widths
and their heights are called the transition strengths.
The experimental nuclear physicists have collected vast amounts of
data concerning the excitation spectra of various nuclei such as those
shown on Figure 1.1 (Garg et al., 1964, where a detailed description
of the experimental work on thorium and uranium energy levels is
given; Rosen et al., 1960; Camarda et al., 1973; Liou et al., 1972a, b).
The ground state and low lying excited states have been impressively
1
2 1. Introduction
1 j j *5*Th C/n)=885 b/otom
I I ,i I I. Ii!l»l I i I ! i I M il
|| ! i i "*U C'/n)» 47β b/otom
Il n il i ll i il > i ί u < »I
400 700 «000
ENERGY (eV)
FlG. 1.1. Slow neutron resonance cross sections on thorium 232 and uranium 238
nuclei. Reprinted with permission from The American Physical Society, Rahn et al.
Neutron resonance spectroscopy, X, Phys. Rev. C, 6, 1854 1869 (1972).
explained in terms of an independent particle model where the nucléons
are supposed to move freely in an average potential well (Mayer and
Jensen, 1955; Kisslinger and Sorenson, 1960). As the excitation energy
increases, more and more nucléons are thrown out of the main body of
the nucleus, and the approximation of replacing their complicated inter-
actions with an average potential becomes more and more inaccurate.
At still higher excitations the nuclear states are so dense and the in-
termixing is so strong that it is a hopeless task to try to explain the
individual states; but when the complications increase beyond a certain
point the situation becomes hopeful again, for we are no longer required
to explain the characteristics of every individual state but only their
average properties, which is much simpler.
The average behavior of the various energy levels is of prime impor-
tance in the study of nuclear reactions. In fact, nuclear reactions may
be put into two major classes—fast and slow. In the first case a typical
reaction time is of the order of the time taken by the incident nucléon
to pass through the nucleus. The wavelength of the incident nucléon is
much smaller than the nuclear dimensions, and the time it spends inside
the nucleus is so short that it interacts with only a few nucléons inside
the nucleus. A typical example is the head-on collision with one nucléon
in which the incident nucléon hits and ejects a nucléon, thus giving it
almost all its momentum and energy. Consequently, in such cases the